Given a Polish dynamical system \((X,T)\) , we say that \(\mu \in \operatorname{\mathcal {M}}(X)\)  is
\(T\) -invariant if \(\mu (T^{-1}(A))=\mu (A)\)  for every Borel set \(A \subseteq X\) .
We let \(\operatorname{\mathcal {M}_{\mathit {T}} }(X)\)  be the set of \(T\) -invariant Borel probability measures on \(X\) .
If \(X\)  is compact, then \(\operatorname{\mathcal {M}_{\mathit {T}} }(X)\)  must be nonempty, but this is no longer true if \(X\)  is just Polish, even if \(T\)  exhibits non-trivial recurrence (see [1]}).
A measure \(\mu \in \operatorname{\mathcal {M}_{\mathit {T}} }(X)\)  is ergodic if for every Borel set \(A\subseteq X\)  satisfying \(T^{-1}(A)\subseteq A\)  it holds that \(\mu (A)\)  is either 0 or 1. We write \(\operatorname{\operatorname{\mathcal {M}}_{\mathit {T}}^e}(X)\)  for the set of \(T\) -invariant ergodic measures. The set \(\operatorname{\mathcal {M}_{\mathit {T}} }(X)\)  is always convex, so \(\operatorname{\operatorname{\mathcal {M}}_{\mathit {T}}^e}(X)\)  is always non-empty if \(\operatorname{\mathcal {M}_{\mathit {T}} }(X)\ne \emptyset \) .
